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BV-operator

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

The concept of BV-operator has two different but (somewhat subtly) related meanings:

  1. In perturbative quantum field theory (discussed below) the BV-operator or BV-Laplacian (Batalin-Vilkovisky 81) is essentially the difference between the action of the BV differential on the algebra of observables before and after quantization of the free field theory around which the perturbative quantization is considered; it is a quantum correction iΔ BVi \hbar \Delta_{BV} (of order of Planck's constant \hbar) to the BV differential s BV={S,.}s_{BV} = \{-S',-.\}. Where the condition ({S,}) 2=0\left(\{-S',-\}\right)^2 = 0 for the BV-differential to be a differential is called the “master equation” in BV-BRST theory, the quantum corrected version ({S,}+iΔ) 2=0\left( \{-S',-\} + i \hbar \Delta\right)^2 = 0 is called the quantum master equation.

  2. In higher algebra, under the identification of a BV-algebra with the chain homology of a E2-algebra, the BV-operator corresponds to the operation of rotating a little disk around.

For the relation between the two see relation between BV and BD.

In perturbative quantum field theory

In perturbative quantum field theory the BV-operator iΔi \hbar \Delta may be understood intuitively as reflecting the contribution of the Gaussian measure in the path integral of the free field theory around which the perturbative quantization takes place. This intuition may be made precise for finite-dimensional toy path integrals. This we discuss in:

In the rigorous construction of relativistic perturbative quantum field theory via causal perturbation theory/perturbative AQFT there is a rigorous incarnation of the BV-operator (Fredenhagen-Rejzner 11b, section 2, Rejzner 11, section 5.1.2): The would-be path integral is reflected in the perturbative S-matrix, hence in the time ordered products, and the BV-operator on regular polynomial observables is the difference between the classical BV-differential and its conjugation into the time-ordered products (def. 2 below). This we discuss in

For finite-dimensional toy path integrals

If XX is a finite dimensional closed oriented smooth manifold, then integration of differential forms of top degree over XX may be identified with sending such differential forms to their image in the de Rham cohomology of XX.

This may be slightly reformulated: Fixing a volume form μ\mu on XX it induces by contraction with vector fields degreewise a linear isomorphism

μ:Γ X( nT *X)Γ X( dim(X)nTX) \mu \;\colon\; \Gamma_X(\wedge^n T^\ast X) \overset{\simeq}{\longrightarrow} \Gamma_X(\wedge^{dim(X)-n} T X)

between the spaces of differential n-forms on XX and the space of multivector fields on XX of degree ndim(X)n-dim(X). Under this isomorphism the de Rham differential induces a differential on multivector fields:

Δ BVμd dRμ 1 \Delta_{BV} \;\coloneqq\; \mu \circ d_{dR} \circ \mu^{-1}

This Δ BV\Delta_{BV} is the BV-operator in this simple situation.

The above statement about integration now translates into saying that for ff any smooth function on XX, then its integration of differential forms fμ\int f \mu may be identified with the image of ff in the chain homology of the BV-operator Δ BV\Delta_{BV}.

If one thinks of XX as a space of field configurations and of f=exp(iS)f = \exp(i \hbar S) as an exponentiated action functional, the one may think of this integral exp(iS)μ\int \exp(i \hbar S) \mu as the finite-dimensional toy version of a path integral.

While this is in general not defined in the actual non-finite dimensional situations in field theory, the above re-formulation in terms of the chain homology of a BV-operator does make sense whenever an appropriate kind of differential is given. One may then try to axiomatize which chain complexes qualify as BV-complex and try to interpret their chain homology as computing perturbative path integrals.

For more on this perspective see at BV-BRST formalism the section Quantum BV as homological (path-)integration

In causal perturbation theory

For more context for the following see at A first idea of quantum field theory the chapter Free quantum fields.

Background

Recall the following context

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory admitting a gauge fixing, and let LL BRST\mathbf{L}' - \mathbf{L}'_{BRST} be its BV-BRST Lagrangian density after gauge fixing (this def.), so that the gauge-fixed local BV-BRST differential is given by the local antibracket as

s={L+L BRST,} s' \;=\; \left\{ -\mathbf{L}' + \mathbf{L}'_{BRST}, - \right\}

The corresponding global BV-BRST differential on regular polynomial observables is (this def.)

(1){S+S BRST,}{τ ΣL(x)+τ ΣL BRST(x),}:PolyObs(E) regPolyObs(E) reg. \left\{ -S' + S'_{BRST} \;,\; -\right\} \;\coloneqq\; \left\{ -\tau_\Sigma\mathbf{L}'(x) + \tau_\Sigma\mathbf{L}'_{BRST}(x), - \right\} \;\colon\; PolyObs(E)_{reg} \longrightarrow PolyObs(E)_{reg} \,.

By definition of gauge fixing (this def.), the Euler-Lagrange equation of motion for L\mathbf{L}' are Green hyperbolic and hence have a causal propagator Deta=Δ +Δ \Deta = \Delta_+ - \Delta_- and admit a compatible Wightman propagator Δ H=i2(Δ +Δ )+H\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H and the corresponding Feynman propagator Δ Fi2(Δ ++Δ )+H\Delta_F \coloneqq \tfrac{i}{2}(\Delta_+ + \Delta_-) + H. The star products with respect to these (this def.) on regular polynomial observables

H, F:PolyObs(E) reg[[]]PolyObs(E) reg[[]] \star_H, \star_F \;\colon\; PolyObs(E)_{reg}[ [\hbar] ] \longrightarrow PolyObs(E)_{reg}[ [\hbar] ]

are, respectively, the Wick algebra product (operator product, see this def.)

A 1A 2A 1 HA 2(()())exp(Σ×ΣΔ H(x,y) abδδΦ a(x)δδΦ b(y)dvol Σ(x)dvol Σ(y))(A 1A 2) A_1 A_2 \;\coloneqq\; A_1 \star_H A_2 \;\coloneqq\; ((-) \cdot (-)) \circ \exp\left( \hbar \underset{\Sigma \times \Sigma}{\int} \Delta_{H}(x,y)^{a b} \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \, dvol_\Sigma(x) \, dvol_\Sigma(y) \right) (A_1 \otimes A_2)

and the time-ordered product (see again this def.)

T(A 1A 2)A 1 FA 2(()())exp(Σ×ΣΔ F(x,y) abδδΦ a(x)δδΦ b(y)dvol Σ(x)dvol Σ(y))(A 1A 2), T(A_1 A_2) \;\coloneqq\; A_1 \star_F A_2 \;\coloneqq\; ((-) \cdot (-)) \circ \exp\left( \hbar \underset{\Sigma \times \Sigma}{\int} \Delta_{F}(x,y)^{a b} \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \, dvol_\Sigma(x)\, dvol_\Sigma(y) \right) (A_1 \otimes A_2) \,,

Since the Feynman propagator is symmetric (this prop.), the latter time ordered product on regular polynomial observables is isomorphic (via this prop.) to the pointwise product, via

(2)𝒯Aexp(12ΣΔ F(x,y) abδ 2δΦ a(x)δΦ b(y))A \mathcal{T}A \;\coloneqq\; \exp\left( \tfrac{1}{2} \hbar \underset{\Sigma}{\int} \Delta_F(x,y)^{a b} \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)} \right) A

as

A 1 FA 2=𝒯(𝒯 1A 1𝒯 1A 2) A_1 \star_{F} A_2 \;=\; \mathcal{T}( \mathcal{T}^{-1}A_1 \cdot \mathcal{T}^{-1}A_2 )

(this prop.).

The BV-Operator

Definition

(time-ordered antibracket)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density L+L BRST-\mathbf{L}' + \mathbf{L}'_{BRST} (this def.) on a graded BV-BRST field bundle E BV-BRSTT *[1] Σ,inf(E× Σ𝒢[1]× ΣA× ΣA[1])E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1]).

Then the time-ordered global antibracket on regular polynomial observables

PolyObs(E BV-BRST) reg[[]]PolyObs(E BV-BRST) reg[[]]{,} 𝒯PolyObs(E BV-BRST) reg[[]] PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \otimes PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \overset{\{-,-\}_{\mathcal{T}}}{\longrightarrow} PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]

is the conjugation of the global antibracket (this def.) by the time-ordering operator 𝒯\mathcal{T} (from this prop.):

{,} 𝒯𝒯({𝒯 1(),𝒯 1()}) \{-,-\}_{\mathcal{T}} \;\coloneqq\; \mathcal{T}\left(\left\{ \mathcal{T}^{-1}(-), \mathcal{T}^{-1}(-)\right\}\right)

hence

PolyObs(E BV-BRST) reg[[]]PolyObs(E BV-BRST) reg[[]] {,} PolyObs(E BV-BRST) reg[[]] 𝒯 𝒯 PolyObs(E BV-BRST) reg[[]]PolyObs(E BV-BRST) reg[[]] {,} 𝒯 PolyObs(E BV-BRST) reg[[]] \array{ PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \otimes PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] &\overset{\{-,-\}}{\longrightarrow}& PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \\ {}^{\mathllap{\mathcal{T}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mathcal{T}}}_\simeq \\ PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \otimes PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] &\overset{ \{-,-\}_{\mathcal{T}} }{\longrightarrow}& PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] }

(Fredenhagen-Rejzner 11, (27), Rejzner 11, (5.14))

Proposition

(time-ordered antibracket with gauge fixed action functional)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density L+L BRST-\mathbf{L}' + \mathbf{L}'_{BRST} (this def.) on a graded BV-BRST field bundle E BV-BRSTT *[1] Σ,inf(E× Σ𝒢[1]× ΣA× ΣA[1])E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1]).

Then the time-ordered antibracket (def. 1) with the gauge fixed BV-action functional S-S' (def. \ref{ComplexBVBRSTGlobal}) equals the conjugation of the global BV-differential with the isomorphism 𝒯\mathcal{T} from the pointwise to the time-ordered product of observables (from this prop.)

{S,} 𝒯=𝒯{S,}𝒯 1, \{-S',-\}_{\mathcal{T}} \;=\; \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1} \,,

hence

PolyObs(E BV-BRST) reg[[]] {S,} PoyObs(E BV-BRST) reg[[]] 𝒯 𝒯 PolyObs(E BV-BRST) reg[[]] {S,} 𝒯 PoyObs(E BV-BRST) reg[[]] \array{ PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] &\overset{ \{-S',-\} }{\longrightarrow}& PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \\ {}^{\mathllap{\mathcal{T}}}\downarrow && \downarrow^{\mathrlap{\mathcal{T}}} \\ PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] &\overset{ \{-S',-\}_{\mathcal{T}} }{\longrightarrow}& PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] }
Proof

By the assumption that (E,L)(E,\mathbf{L}) is a free field theory its Euler-Lagrange equations are linear in the fields, and hence SS' is quadratic in the fields. This means that

𝒯 1S=S+const, \mathcal{T}^{-1}S' = S' + const \,,

where the second term on the right is independent of the fields, and hence that

{𝒯 1(S),}={S,}. \{\mathcal{T}^{-1}(-S'),-\} = \{-S', - \} \,.

This implies the claim:

{S,} 𝒯 𝒯({𝒯 1(S),𝒯 1()}) =𝒯({S,𝒯 1()}) =𝒯{S,}𝒯 1. \begin{aligned} \{-S',-\}_{\mathcal{T}} & \coloneqq \mathcal{T}\left(\{ \mathcal{T}^{-1}(-S'), \mathcal{T}^{-1}(-) \}\right) \\ & = \mathcal{T}\left(\{ -S', \mathcal{T}^{-1}(-) \}\right) \\ & = \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1} \,. \end{aligned}
Definition

(BV-operator for gauge fixed free Lagrangian field theory)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density L+L BRST-\mathbf{L}' + \mathbf{L}'_{BRST} (this def.) on a graded BV-BRST field bundle E BV-BRSTT *[1] Σ,inf(E× Σ𝒢[1]× ΣA× ΣA[1])E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1]) and with corresponding gauge-fixed global BV-BRST differential on graded regular polynomial observables

{S+S BRST,}:PolyObs(E BV-BRST) reg[[]]PolyObs(E BV-BRST) reg[[]] \{-S' + S'_{BRST}, -\} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]

(this def.).

Then the corresponding BV-operator

Δ BV:PolyObs(E BV-BRST) reg[[]]PolyObs(E BV-BRST) reg[[]] \Delta_{BV} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]

on regular polynomial observables is, up to a factor of ii \hbar, the difference between the free component {S,}\{-S',-\} of the gauge fixed global BV differential its time-ordered version (def. 1)

Δ BV1i({S,} 𝒯{S,()}), \Delta_{BV} \;\coloneqq\; \tfrac{1}{i \hbar} \left( \left\{ -S',- \right\}_{\mathcal{T}} - \left\{ -S',(-) \right\} \right) \,,

hence

(3){S,} 𝒯={S,}+iΔ BV. \{-S',-\}_{\mathcal{T}} \;=\; \{-S',-\} + i \hbar \Delta_{BV} \,.
Proposition

(BV-operator in components)

If the field bundles of all fields, ghost fields and auxiliary fields are trivial vector bundles, with field/ghost-field/auxiliary-field coordinates collectively denoted (ϕ A)(\phi^A) then the BV-operator Δ BV\Delta_{BV} from prop. 2 is given explicitly by

Δ BV=a(1) deg(Φ A)ΣδδΦ A(x)δδΦ A (y)dvol Σ \Delta_{BV} \;=\; \underset{a}{\sum} (-1)^{deg(\Phi^A)} \underset{\Sigma}{\int} \frac{\delta}{\delta \Phi^A(x)} \frac{\delta}{\delta \Phi^{\ddagger}_A(y)} dvol_\Sigma

Since this formula exhibits a graded Laplace operator, the BV-operator is also called the BV-Laplace operator or BV-Laplacian, for short.

(Fredenhagen-Rejzner 11, (29), Rejzner 11, (5.20))

Proof

By prop. 1 we have equivalently

iΔ BV=𝒯{S,}𝒯 1{S,} i \hbar \Delta_{BV} \;=\; \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1} \,-\, \{-S',-\}

and by this example the second term on the right is

{S,} =Σj (Φ) *(δ ELLδϕ A)(x)δδΦ A (x)dvol Σ(x) =a(1) deg(ϕ A)(PΦ) A(x)δδΦ A (x)dvol Σ(x) \begin{aligned} \left\{ -S', -\right\} & = \underset{\Sigma}{\int} j^{\infty}\left(\mathbf{\Phi}\right)^\ast \left( \frac{\overset{\leftarrow}{\delta}_{EL} L}{\delta \phi^A} \right)(x) \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \, dvol_\Sigma(x) \\ & = \underset{a}{\sum} (-1)^{deg(\phi^A)} \underset{}{\int} (P\mathbf{\Phi})_A(x) \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \, dvol_\Sigma(x) \end{aligned}

With this we compute as follows:

(4){S,} 𝒯 =𝒯{S,}𝒯 1 =exp([12Δ F(δδΦ,δδΦ),])({S,}) ={S,}+[12Δ F(δδΦ,δδΦ),{S,}]+ 2(...)=0 =+{S,} =+[12Σ×ΣΔ F AB(x,y)δ 2δΦ A(x)δΦ B(y)dvol Σ(x)dvol Σ(y),a(1) deg(ϕ A)Σ(PΦ) A(x)δδΦ A (x)dvol Σ(x)] ={S,} =+A(1) deg(ϕ A)Σ×ΣP xΔ F(x,y)=iδ(xy)δδΦ A(x)δδΦ A (y)dvol Σ(x)dvol Σ(y) ={S,}+iA(1) deg(ϕ A)ΣδδΦ A(x)δδΦ A (x)dvol Σ(x) \begin{aligned} \{-S',-\}_{\mathcal{T}} & = \mathcal{T} \circ \left\{ -S,-\right\} \circ \mathcal{T}^{-1} \\ & = \exp\left( \left[ \hbar \tfrac{1}{2} \Delta_F \left( \frac{\delta}{\delta \mathbf{\Phi}}, \frac{\delta}{\delta \mathbf{\Phi}} \right) \,,\, - \right] \right) \left( \{-S',-\} \right) \\ & = \{-S',-\} + \left[ \hbar \tfrac{1}{2} \Delta_F \left( \frac{\delta}{\delta \mathbf{\Phi}}, \frac{\delta}{\delta \mathbf{\Phi}} \right) \,, \{-S',-\} \right] + \underset{ = 0 }{\underbrace{\hbar^2(...)}} \\ & = \phantom{+} \left\{ -S' , -\right\} \\ & \phantom{=} + \left[ \tfrac{1}{2}\hbar \underset{\Sigma \times \Sigma}{\int} \Delta_F^{A B}(x,y) \frac{\delta^2}{\delta \mathbf{\Phi}^A(x) \delta \mathbf{\Phi}^B(y)} \, dvol_\Sigma(x) \, dvol_\Sigma(y) \;,\; \underset{a}{\sum} (-1)^{deg(\phi^A)} \underset{\Sigma}{\int} (P\mathbf{\Phi})_A(x) \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \, dvol_\Sigma(x) \right] \\ & = \left\{ -S', -\right\} \\ & \phantom{=} + \underset{A}{\sum} (-1)^{deg(\phi^A)} \underset{\Sigma \times \Sigma}{\int} \underset{ = i \delta(x-y) }{\underbrace{P_x \Delta_F(x,y)}} \frac{\delta}{\delta \mathbf{\Phi}^A(x)} \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(y)} \, dvol_\Sigma(x) \, dvol_\Sigma(y) \\ & = \left\{ -S', -\right\} + i \hbar \underset{A}{\sum} (-1)^{deg(\phi^A)} \underset{\Sigma}{\int} \frac{\delta}{\delta \mathbf{\Phi}^A(x)} \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \, dvol_\Sigma(x) \end{aligned}

Here we used

  1. under the first brace that by assumption of a free field theory, {S,}\{-S',-\} is linear in the fields, so that the first commutator with the Feynman propagator is independent of the fields, and hence all the higher commutators vanish;

  2. under the second brace that the Feynman propagator is +i+i times the Green function for the Green hyperbolic Euler-Lagrange equations of motion (this cor.).

Relation to time-ordered antibracket

Proposition

(global antibracket exhibits failure of BV-operator to be a derivation)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density L+L BRST-\mathbf{L}' + \mathbf{L}'_{BRST} (this def.) on a graded BV-BRST field bundle E BV-BRSTT *[1] Σ,inf(E× Σ𝒢[1]× ΣA× ΣA[1])E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])

The BV-operator Δ BV\Delta_{BV} (def. 2) and the global antibracket {,}\{-,-\} (this def.) satisfy for all polynomial observables (this def.) A 1,A 2PolyObs(E BV-BRST)[[]]A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})[ [\hbar] ] the relation

(5){A 1,A 2}=(1) deg(A 2)Δ BV(A 1A 2)(1) deg(A 2)Δ BV(A 1)A 2A 1Δ BV(A 2) \{A_1, A_2\} \;=\; (-1)^{deg(A_2)} \, \Delta_{BV}(A_1 \cdot A_2) - (-1)^{deg(A_2)} \, \Delta_{BV}(A_1) \cdot A_2 - A_1 \cdot \Delta_{BV}(A_2)

for ()()(-) \cdot (-) the pointwise product of observables (def. \ref{Observable}).

Moreover, it commutes on regular polynomial observables with the time-ordering operator 𝒯\mathcal{T} (this prop.)

Δ BV𝒯=𝒯Δ BVAAAonPolyObs(E BV-BRST) reg[[]] \Delta_{BV} \circ \mathcal{T} = \mathcal{T} \circ \Delta_{BV} \phantom{AAA} \text{on} \,\, PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]

and hence satisfies the analogue of relation (5) also for the time-ordered antibracket {,} 𝒯\{-,-\}_{\mathcal{T}} (def. 1) and the time-ordered product F\star_F on regular polynomial observables

{A 1,A 2} 𝒯=(1) deg(A 2)Δ BV(A 1 FA 2)(1) deg(A 2)Δ BV(A 1) FA 2A 1 FΔ BV(A 2). \{A_1, A_2\}_{\mathcal{T}} \;=\; (-1)^{deg(A_2)} \, \Delta_{BV}(A_1 \star_F A_2) - (-1)^{deg(A_2)} \Delta_{BV}(A_1) \star_F A_2 - A_1 \star_F \Delta_{BV}(A_2) \,.

(e.g. Henneaux-Teitelboim 92, (15.105d))

Proof

With prop. 2 the first statement is a graded version of the analogous relation for an ordinary Laplace operator Δg ab a b\Delta \coloneqq g^{a b} \partial_a \partial_b acting on smooth functions on Cartesian space, which on smooth functions f,gf,g satisfies

Δ(fg)=(f,g)Δ(f)gfΔ(g), \Delta(f \cdot g) \;=\; (\nabla f, \nabla g) - \Delta(f) g - f \Delta(g) \,,

by the product law for differentiation, where now f(g ab bf)\nabla f \coloneqq (g^{a b} \partial_b f) is the gradient and (v,w)g abv awb(v,w) \coloneqq g_{a b} v^a w b the inner product. Here one just needs to carefully record the relative signs that appear.

That the BV-operator commutes with the time-ordering operator is clear from the fact that both of these are given by partial functional derivatives with constant coefficients. This immediately implies the last statement from the first.

Example

(BV-operator on time-ordered exponentials)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density L+L BRST-\mathbf{L}' + \mathbf{L}'_{BRST} (this def.) on a graded BV-BRST field bundle E BV-BRSTT *[1] Σ,inf(E× Σ𝒢[1]× ΣA× ΣA[1])E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1]).

Let moreover VPolyObs(E BV-BRST) reg,deg=0[[]]V \in PolyObs(E_{\text{BV-BRST}})_{reg, deg = 0}[ [\hbar] ] be a regular polynomial observable (def. \ref{PolynomialObservables}) of degree zero. Then the application of the BV-operator Δ BV\Delta_{BV} (def. 2) to the time-ordered exponential exp 𝒯(V)\exp_{\mathcal{T}}(V) (this example) is the time-ordered product of the time-ordered exponential with the sum of Δ BV(V)\Delta_{BV}(V) and the global time-ordered antibracket 12{V,V} 𝒯\tfrac{1}{2}\{V,V\}_{\mathcal{T}} (def. 1) of VV with itself:

Δ BV(exp 𝒯(V))=(Δ BV(V)+12{V,V} 𝒯) Fexp 𝒯(V) \Delta_{BV} \left( \exp_{\mathcal{T}}(V) \right) \;=\; \left( \Delta_{BV}(V) + \tfrac{1}{2}\{V,V\}_{\mathcal{T}} \right) \star_F \exp_{\mathcal{T}}(V)
Proof

By prop. 3 Δ BV\Delta_{BV} acts as a derivation on the time-ordered product up to a correction given by the antibracket of the two factors. This yields the result by the usual combinatorics of exponentials.

Δ BV(1+V+12V FV+) =Δ BV(V)+12(Δ BV(V) FV+V FΔ BV(V))+12{V,V} 𝒯+ =Δ BV(V)+Δ BV(V) FV+12{V,V} 𝒯+ \begin{aligned} \Delta_{BV} \left( 1 + V + \tfrac{1}{2}V \star_F V + \cdots \right) & = \Delta_{BV}(V) + \tfrac{1}{2}\left( \Delta_{BV}(V) \star_F V + V \star_F \Delta_{BV}(V) \right) + \tfrac{1}{2}\{V,V\}_{\mathcal{T}} + \cdots \\ & = \Delta_{BV}(V) + \Delta_{BV}(V) \star_F V + \tfrac{1}{2}\{V,V\}_{\mathcal{T}} + \cdots \end{aligned}

Schwinger-Dyson equation

A special case of the general occurence of the BV-operator is the following important property of on-shell time-ordered products:

Proposition

(Schwinger-Dyson equation)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density L+L BRST-\mathbf{L}' + \mathbf{L}'_{BRST} (this def.) on a graded BV-BRST field bundle E BV-BRSTT *[1] Σ,inf(E× Σ𝒢[1]× ΣA× ΣA[1])E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1]).

Let

(6)AΣA a(x)Φ a (x)dvol Σ(x)PolyObs reg(E BV-BRST) A \;\coloneqq\; \underset{\Sigma}{\int} A^a(x) \cdot \mathbf{\Phi}^\ddagger_a(x) \, dvol_\Sigma(x) \;\in\; PolyObs_{reg}(E_{\text{BV-BRST}})

be an off-shell regular polynomial observable which is linear in the antifield field observables Φ \mathbf{\Phi}^\ddagger. Then

(7)𝒯 ±1(ΣδSδΦ a(x)A a(x)dvol Σ(x))=±i𝒯 ±(ΣδA a(x)δΦ a(x)dvol Σ(x))APolyObs reg(E BV-BRST,L)on-shell. \mathcal{T}^{\pm 1} \left( \underset{\Sigma}{\int} \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \cdot A^a(x) \, dvol_\Sigma(x) \right) \;=\; \pm i \hbar \, \mathcal{T}^{\pm} \left( \underset{\Sigma}{\int} \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)} \, dvol_\Sigma(x) \right) \phantom{A} \in \underset{ \text{on-shell} }{ \underbrace{ PolyObs_{reg}(E_{\text{BV-BRST}}, \mathbf{L'}) }} \,.

This is called the Schwinger-Dyson equation.

The proof below is due to (Rejzner 16, remark 7.7), following the informal traditional argument (Henneaux-Teitelboim 92, (15.108b)).

Proof

Applying the inverse time-ordering map 𝒯 1\mathcal{T}^{-1} (this prop.) to equation (3) applied to AA yields

(8)𝒯 1{S,A}𝒯 1ΣδSδΦ a(x)A a(x)dvol Σ(x)=i𝒯 1Δ BV(A)i𝒯 1ΣδA a(x)δΦ a(x)dvol Σ+𝒯 1{S,A} 𝒯{S,𝒯 1(A)} \underset{ \mathcal{T}^{-1} \underset{\Sigma}{\int} \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \cdot A^a(x) dvol_\Sigma(x) }{ \underbrace{ \mathcal{T}^{-1}\left\{ -S', A\right\} } } \;=\; - \underset{ i \hbar \mathcal{T}^{-1} \underset{\Sigma}{\int} \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)} \, dvol_\Sigma }{ \underbrace{ i \hbar \mathcal{T}^{-1}\Delta_{BV}(A) } } + \underset{ \{-S',\mathcal{T}^{-1}(A)\} }{ \underbrace{ \mathcal{T}^{-1}\left\{ -S',A\right\}_{\mathcal{T}} } }

where we have identified the terms under the braces by 1) the component expression for the BV-differential {S,}\{-S',-\} from this prop, 2) prop. 2 and 3) prop. 1.

The last term is manifestly in the image of the BV-differential {S,}\{-S',-\} and hence vanishes when passing to on-shell observables along the isomorphism (this equation)

(9)PolyObs(E BV-BRST,L)on-shellPolyObs(E BV-BRST) def(af=0)off-shell/im({S,}) \underset{ \text{on-shell} }{ \underbrace{ PolyObs(E_{\text{BV-BRST}}, \mathbf{L}') }} \;\simeq\; \underset{ \text{off-shell} }{ \underbrace{ PolyObs(E_{\text{BV-BRST}})_{def(af = 0)} }}/im(\{-S',-\})

(by this example.).

The same argument with the replacement 𝒯𝒯 1\mathcal{T} \leftrightarrow \mathcal{T}^{-1} throughout yields the other version of the equation (with time-ordering instead of reverse time ordering and the sign of the \hbar-term reversed).

Remark

(“Schwinger-Dyson operator”)

The proof of the Schwinger-Dyson equation in prop. 4 shows that, up to time-ordering, the Schwinger-Dyson equation is the on-shell vanishing of the “quantized” BV-differential (3)

{S,} 𝒯={S,}+iΔ BV, \{-S',-\}_{\mathcal{T}} \;=\; \{-S', -\} \,+\, i \hbar \, \Delta_{BV} \,,

where the BV-operator is the quantum correction of order \hbar. Therefore this is also called the Schwinger-Dyson operator (Henneaux-Teitelboim 92, (15.111)).

Remark

(distributional Schwinger-Dyson equation)

Often the Schwinger-Dyson equation (prop. 4) is displayed before spacetime-smearing of field observables in terms of operator products of operator-valued distributions, taking the observable AA in (6) to be

A a(x)δ(xx 0)δ a 0 aΦ a 1(x 1)Φ a n(x n). A^a(x) \;\coloneqq\; \delta(x-x_0) \delta^a_{a_0} \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n) \,.

This choice makes (7) become the distributional Schwinger-Dyson equation

T(δSδΦ a 0(x 0)Φ a 1(x 1)Φ a n(x n)) =on-shellikT(Φ a 1(x 1)Φ a k1(x k1)δ(x 0x k)δ a k a 0Φ a k+1(x k+1)Φ a n(x m)) \begin{aligned} & T \left( \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)} \cdot \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n) \right) \\ & \underset{\text{on-shell}}{=} - i \hbar \underset{k}{\sum} T \left( \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_{k-1}}(x_{k-1}) \cdot \delta(x_0 - x_k) \delta^{a_0}_{a_k} \cdot \mathbf{\Phi}^{a_{k+1}}(x_{k+1}) \cdots \mathbf{\Phi}^{a_n}(x_m) \right) \end{aligned}

(e.q. Dermisek 09).

In particular this means that if (x 0,a 0)(x k,a k)(x_0,a_0) \neq (x_k, a_k) for all k{1,,n}k \in \{1,\cdots ,n\} then

T(δSδΦ a 0(x 0)Φ a 1(x 1)Φ a n(x n))=0AAAon-shell T \left( \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)} \cdot \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n) \right) \;=\; 0 \phantom{AAA} \text{on-shell}

Since by the principle of extremal action (this prop.) the equation

δSδΦ a 0(x 0)=0 \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)} \;=\; 0

is the Euler-Lagrange equation of motion (for the classical field theory) “at x 0x_0”, this may be interpreted as saying that the classical equations of motion for fields at x 0x_0 still hold for time-ordered quantum expectation values, as long as all other observables are evaluated away from x 0x_0; while if observables do coincide at x 0x_0 then there is a correction measured by the BV-operator.

Remark

(the “quantum shell”)

Beware that, superficially, it might seem that in equation (8) not only the term {S,𝒯 1(A)}\{-S',\mathcal{T}^{-1}(A)\} on the right vanishes on-shell, but also the term 𝒯 1{S,A}\mathcal{T}^{-1}\left\{ -S', A\right\} on the left, since the latter is the image under the linear map 𝒯 1\mathcal{T}^{-1} of an observable that vanishes on-shell.

To sort this out, notice that the isomorphism (9) tells us that the observables that vanish when passing from off-shell to on-shell observables are precisely those in the ideal generated by the image of {S,()}\{-S',(-)\}. But while 𝒯 1\mathcal{T}^{-1} is an isomorphism on (regular off-shell observables), it need not (and in general does not) preserve this ideal! Hence 𝒯 1({S,A})\mathcal{T}^{-1}(\{-S',A\}) need not (and in general is not) an element of that ideal, and this is why it remains when passing to the algebra of on-shell observables and thus makes its crucial appearance in the Schwinger-Dyson equation.

Quantum master equation

Passing from free field theory to perturbative quantization of interacting field theory, the above BV-operator of the underying free field theory appears as a quantum correction to the “classical master equation” which expresses the nilpotency of the BV-differential. This “master equation” corrected by the BV-operator is called the quantum master equation. See this prop..

References

The concept originates with

Traditional review includes

The understanding of the BV-operator in the rigorous formulation of relativistic perturbative quantum field theory via causal perturbation theory/perturbative AQFT is due to

surveyed in

See at BV-formalism for more references.

Last revised on March 12, 2018 at 10:13:33. See the history of this page for a list of all contributions to it.